Slide-rule.



No. 753,840. PATENTEDMAR. 8, 1904.

C. G. BARTH, H.. L GANTT 6r F. TAYLOR. l I SLIDE RULE. l

l urmcwmm Hmm Nov. 2o. 1901.

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ENTWIS:

No. 753,840. PATENTED MAR. 8, 1904. C. G. BARTH, H. L. GANTT' da F. W. TAYLOR.

SLIDE RULE.

APPLIGATION HLBD Nov. 2o. 1901.

5 SHEETS-SHEET 2.

NO MODEL.

'dal' /N VEN THS I .7M

/fww /M/ WITNESSES: femwmmg 110.753,84@ RATRNTRD MAR. a, 1904. c; G'. BARTH, R. L GANTT R P. W. TAYLOR.

SLIDE RULR. L

APPLIGATION FILED NOV. 20, 1901l I0 HODBL. 5 SHEETS-SHEET 3.

PATENTED MAR. 8, 1904.

No. 753,840. n

G. G. BARTH, H. L. GANTT & P. W. TAYLOR.

SLIDE RULE.

APPLIUATIOH FILED NOV. 20, 1901.

5 SHEETS-SHEET 4.

BER 0r T NUM 2 sans- WQNESSS: v lgvfETons:

1660 Vm ML l w: Y H6324' v6 o@ @MX/C M7 Y 'PATBNTLD MAR. s, 1904.

SLIDE RULE.

5 SHEETS-sum 5.

-INVENTOB$.' fm M.

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APPLICATION FILED NOV. 20, 1901.

E ow JA. 2 2 QE v C. G. BARTH, H. L. GANTT & F. W. TAYLOR.

SX w m n v u n u .2. 1 2 m3 d eN E 2mm b d ,ma ha m6 *e n d .m-.o .o 5.o wad d No MODEL.

WITNESSES: w/MM Wk-f4" UNITED. STATES Patented march s, 1904..l

PATENT OFFICE.

CARL G. BARTH, OF BETHLEHEM, AND HENRY L. GANTT AND FREDERICK W. TAYLOR, OF SOUTH BETHLEHEM, PENNSYLVANIA.

SLIDE-RULE.

SPECIFICATION forming part of Letters Patent No. 753,840, dated March 8, 1904.

Application iai November 29,1901. serial No. 82,998. No miei.)

To all whom it may concern:

Be it known that we, CARL G. BARTH, a resident of Bethlehem, and HENRY L. GANTT and FREDERICK W. TAYLOR, residents of South Bethlehem, in the county of Northampton, State of Pennsylvania, allcitizens of the United States ofAmerica, have invented certain new and useful Improvements in Slide-Rules, of which the following is a true and exact description, reference being had to the accompanying drawings, which form a part thereof.

Our invention'relates to slide-rules.

The chief object of our invention is to enlarge the scope of these useful and time-saving instruments.

Other objects include hereinafter specifically-defined constructive details.

Our still further objects will be best understood from this specification as a whole.

Our invention is addressed to problems which may besaid, in ,mathematical language, to involve more than one equation or set of relations between the variables entering into the problem. Of these variables, in any particular case under consideration, some are known and others, and as many as there are equations in the case to be solved, are unknown.

`Our invention in Vits broad aspect may be said to consist in the construction of a sliderule which gives answer to above classified problems.

In carrying our invention into practice we incorporate upon a slide-rule, and preferably upon attached sections thereof, distinct mechanical embodimentswsuch as logarithmic scales, linear or circular*recognized as means of representing the variable arithmetical values insertable in those formulae or equations which as above set forth are incident to the problems solvable vby our sliderules.

Referencenow being had to the aforesaid. drawings, they will be found to illustrate our invention as follows: 1

. Figure 1 is a plan view of a slide-rule embodying our invention in a single instrument. Fig. 2 is an end elevation of the same on a larger scale. Fig. 3 is an end elevation of a fragment of a modification giving other details of construction in the more purely me- `circular arrangement.

chanical combinations of one of our rules. Fig. 4 is a plan view of another embodiment of our invention, adapted to the solution of a different specific problem from that which is within the scope of the rule shown in Figs. 1 and 2. Fig. 5 is an end elevation, on a a larger scale, of the slide-rule shown in Fig. 4. Figs. 6 and 7 are fragmented plan and side elevations, respectively, of the carrier part of a composite slide seen removed from its base or relatively fixed part of its rule and constitutes an ancillary detailof our invention. Figs. 8 and 9 are plan and side elevations, respectively, of a carried part of said composite slide, the same being typical of an interchangeable lot of similar parts any one of which having embodied upon it a germane logarithmic scale, &c. may by manual insertion be interlockingly engaged and then as a composite slide insertible y for working in its proper guide of a slide-rule. Figs. 10 and 11 are cross-section and end View, respectively, of the composite slide shown in Figs. 6, 7, 8, and 9, wherein broken line 10 10 vdenotes plane of section in Fig. 10. Fig. 12

is a plan view of an embodiment of our invention to solve the same problems as those solved by the instruments shown in Figs. l and 2; but instead of the logarithmic scales being embodied in rectilinear arrangement, as they are in said prior instance, the same are here in So, too, in Fig. 12 the slides of the circular instrument instead of being nested in a base have a disk-like form and are piled in order one upon the other and adapted, as is usual in circular slide-rules, to swing over each other about a common center. Fig. 13 is a cross-section on median line 13 13 of the rule shown in Fig. 12.d Fig. 14 is a plan view of a further modification of the circular ,sort of our slide-rule equipped to solve in a more restricted range the same problem as the slide-rule shown in Figs. 4 and 5. Fig. l5 is a plan view of a further modification of our invention, aimed to Solve problems of the classified sort which involve more unknown variables than are involved in the problems solvable by the slide-rules shown in the prior figures.

As a premise to the descriptonof those of the illustrated slide-rules that display rectilinear scaleswe shall take up the more vpurelymechanical detail of their sliding parts.

in Fig. 1 the slides B B2 B3 are all similarly-dimensioned p ismatic bodies of Z-seC- tion. (See end vie Fig. 2.) rllhe guides, in which they work le gthwise, are Z-slots b b2 b3, sunk in parallel an running along the relatively fixed base A f om end to end of the nconjunction.) rl`his section below the waist b of such slides, (see' slides B B2, Fig. 2, also B31 to B33, Fig. 3, and B41 to B in Fig. 5,) we nd sufficient in average usage for the self-retention to its base of any isolated slide, as slide B3, Figs. 1 and 2; but in the case of such adjoining slides, as the t'wins B B2, Figs. 1,2,whereof the scale-bearing Inargins g g2 should, for easily determining the ,incidents of their scale-lines, come close alongside each other, we preferably contrive that a very small clearance-exaggerated for drafting reasons-at 0'- shall separatetheir upper and otherwise meeting ange's b* b3, and by reason of the closer fit which we form between the opposite an'ges 53 67 of each said Z-shaped slide and said flanges corresponding female parts b3 b3 of' the slots that guide'them'(see Fig. 2) it is contrivedthat each slot and slide f the twin and neighboring workmates B B2, thus laterally restricted to less limits than 'said clearance, shall cooperate to prevent either slide when moving in its guide from dislocating its mate.A The great convenience of this in vadjoining twin slides only need to be stated to be appreciated.

While we prefer the Z-section thus villustrated for adjoining twin scales,when itcomes to'three Iadjoining neighbors in a block, as is the case in Fig. 4f, we contrive, preferably, a U-section for the intermediate slide,-hereB.

.This slide at its waist b3 (see Fig. 5 )being close fitted laterally, but not too snugly for sliding between the guide-faces b '512, is also `cleared at cz c3 from the top ianges b* b3 of its anliing neighbors, whereby said 4neighbors can each`be Worked with absolute independence. Also where by the exigencies of the problems to which our slide-rules are appropriate a number of mutually-flanking slides Vgreater than three is required we have contrived I-shaped symmetrical slides, such as YB31B32 B33 of Fig. 3. These slides, cleared between their upper adjoining flanges-viz., at 033 c32-and laterally snug-fitted to their guides in their narrow waists 633, &c., can be severally manipulated with a corresponding freedom from mutual dislocation.

in the modification Fig. 3 it will be noted that the cleared top flanges of the llsection slides B31, cc., stand free and lying abreast forma requisite flush face A33 for the mechanical embodiment in close juxtaposition of the desired scales.

'ln some cases a rule, more especially such as shown in Figs. 1 2, and 3, 4, may call for scale or scale part different from that with which the instrument is at some time equipped. To this ende-we have devised and combined in a slide-rule a detachable scale-bearing part or parts, viz:

First. f in the relatively fixed part or parts of the rule we do so, preferably as shown in Figs-4, 5, by a detachable block K', here a right parallelopiped formed and fitted to lie flush in a registering sunk chamber K2, and from the face of said block, preferably at one end, countersink an undercut finger-hold K3, by which "hold the user can when desired prize out such block from its chamber, it is obvious such block may be taken as typical of an interchangeable series of blocks, upon which of course any desired and germane scale or scale part could be embodied and of which any one can therefore be put in the exact place of the removed one.

Second. lf it is desired to make a change of scale in a slide, we have devised and combined in a slide-rule the composite slide al-` ready, as to Figs. 6-to 11, partially described in the catalogue of figures. 0f it there is, however, to be added that as a means of initially interlocking the carrier part It with the carried part k2 we preferably provide the same with male and female,l k3, .interlocking 'rabhetst The illustrations Figs. 6 to 12 give a form thereof which we haveV found ecientyarise as to the strength and deflection of helical springs. To this end the formulae shown in label C, Fig. 1, symbolically express the two equations, or sets of relations, between the variables entering into the problems of this rules scope. `The said formulae are thus displayed upon the rules face A3 as aconvenient means'of preventing mistaken selections in the manipulation of the several sliding and fixed parts of the rule. Moreover, in them, and y, the conventional symbols for such purpose designate the two unknown variables.

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Inspection of this rule will also show that every scale (indicated at a', &c.) is for a similar precaution labeled with its own terse title.

We have also indicated by broken line W W' that this rule, which is physically united into a single instrument, may yet be considered as divisible into two sections A A2, respectively. One of these sections is given up to logarithmic scales, (identified by a2 to @2) which mechanically embody the formula for deflection of helical springs. (See label C.) To this formula, as is understood by those skilled in the art of spring-designing, there is a second, to wit, a,formula for load of helical springs. To this latter formula the other, or A2, section of this rule is appropriated, and thereon in Fig. 1 will be seen otherlogarithmic scales, (indicated by as a a 1r/2) by means of which saidsecond formula also becomes a component in the rule. Now of these several logarithmic scales the scales a2 and af (ofthe equation for deiection) are on the fixed parts Ao of their proper section A', and the scales as and a7 are upon the fixed or base portion of A of their section A2 of the slide-rule. 4Upon the face or plan side, Fig. 1, of slide B is the scale as, which scale expresses the number of free coils, a variable subject to ascertainment in any given case, and which (designated'as N) appearsy in the formula for deection, label C. Scale a* is a similar logarithmic expression mechanically embodied on slide B for the modulus of elasticity, another variable term in same formula. Scales t5 and a6 on margins of slide B2 express within their limits other variables of-the same (deflection) and of mean diameter of coil.

formula-viz., that of the torsional stress (See label C.) Of these, a6 is to be distinguished from those preceding it, (a2 to @2) being vthe mei chanical embodiment of one of the unknown This tabulation is introduced sim variables of said formula and which same unknown variable, but in a different logarithmic scale a', also appears on the inside of slide B3 of the aforesaid formulaioiiiload or A2 section' of the rule. 4 Y

In the relatively fixed middle part of the slide-rule between scales a5 and a is plotted the, as it were, split or duplex logarithmic scale a7, a7". This scale mechanically embodies in the rule the other unknown variable of or common to `both aforesaid formulae-viz., diameter of wire. It is also to be noted that distributed along this fixed middle part of the rule and sandwiched between scales a7 -afx (along line W W) is a logarithmix tabulation of Brown & Sharpes wire gage. n p yas an additlonal convenience, so that when (as set forth in the next ensuing example of this rules A1se) the rule having beenfset to the requirements of a given case within its scope and it consequently reveals the answer there may b e' shown as near as may be to said anlswer the nearest sizeof Wire that the Brown & Sharpes gage can with respect to safety afford.

As an example of the duplex function of the rule shown in Figs. 1 and 2 We will suppose that the designer wants to ascertain what mean diameter of coils and of what diameter of wire a helical steel spring of five free coils must be made in order to safely sustain a load of ninety pounds under a deiection of one inch 'when the material .to be used has a 4modulus of elasticity for torsion of eleven million pounds and can work safely under a torsional stress of seventy thousand pounds.

First. vTo satisfy the requirement of load,l we so adjust the slide B3 that the seventy thousand pounds graduation-mark on the scale marked torsional stress coincides with the ninety pounds graduation-mark on the load scale, when We at once have before our eyes all simultaneous values of wire and coil diameters that will satisfy this one requirement.

Second. To satisfy the requirement of deiiection, we so adjust the slide B that the graduation-mark 5 on the scale of number of free coils coincides with the one-inch graduationnmark on the scale of deflection and then also so adjust slide B2 that the seventy thousand pounds graduation-mark on the scale marked torsional stress coincides with the eleven million pounds graduation-mark on the scale marked modulus of elasticity for torsion, when we also at once have before our eyes all possible combinations of coil and wire diameter thatwill satisfy the requirement as to deiiection.

For a person familiar with the instrument the particular values of coil and Wire diameter that at the same time satisfy both requirements are then at once revealed, these being in this particular case a two-inch diameter of coil and 0.4 of an inch diameter of Wire, as more clearly pointed out by the coincident gradua-` tion-marks of this example being made in heavier lines i than the rest.

The lathe slide-rule shown in Figs. 4 and 5 is typical of a class of our slide-rules designed for the purpose of determining under certain conditions the feed and speed at which to run a machine-tool in order to remove the greatest amount of material in the shortest possible time.

As a preliminary to the construction of IOO such an instrument as the slide-rule shown in Figs. 4 and 5 typifies we made extensive study of the relations between the size and shape of a cutting-tool working on various ma- .terials, the depth and feed of the cut, the cutting speed, and the length of time-the tool Which may be embodied in a logarithmic sliderule. We have already constructed such instruments for lathes,planers, and drill-presses, respectively. Of these the particular sliderule illustrated in Figs. 4 and 5 is typical. It is so arranged that it may be used for any lathe whose speed and power combinations fall within its limits. In the first place to this .-protean end the two slides B43 and B44 hereof are of the composite type above described in connection with Figs. 6 to 11, and, secondly, Vthe duplex scale a @47X in the middle of the instrument isalso only, as hereinabove described, a temporarily-inserted one.

It is in this instance that part of the rule vwhich carries the scale of feeds available on the particular lathe for which for the Ytime being the slide-rule, Figs. 4 and 5, is equipped, here a lathe designated for particularization only as No. 43. Lathe No. 43 to be presumed is of the ordinary cone-pulley beltdriven class, its cones having five steps, designated from the largest to the smallest on the cone of itshead-stockby 1, 2, 4 3, 4,and 5.

It also has two combinations of reducing or f 4 back gears between its head-stock cone and its spindle, the greater reduction being designated by "A and the lesser as B. The counter-cone of lathe 43 is itself provided with two different speeds, the slower of which speeds is designated by S and the faster by F So tively fixed the lathe`43 has consequently in all twenty different speed and power com binations, of which number, however, only eighteen, for consideration of space, have been put on the slide-rule, Figs. 4 and 5. So equipped and nomenclatured such a lathe as No. 43 will be recognized and familiar to machine-tool makers.

On the lower or speed section A42 of the Sliderule, Figs. 4 and 5, are logarithmic scales mechanically embodying the law of relation (developed as aforesaid) between the life of tool or the length of time the toolJ will stand up to the work, (scale 0142,) the use or non-use of a cooling-stream of water on the tool, (the broad arrows on scale B42, 'exponented dry and with water, respectively,) the This scale t a and marked not only marginally in decimal and vulgar fractional parts of an inch, but

centrally also by an alphabetical scale of the similarly-marked parts, which in our correlated lathe No. 43 mechanically embody its feed-controlling devices. This law thus embodied upon the slide-rule constitutes one equation of the general problem characteristics of lathe No. 43, the unknown variables or quantities of which are respectively the speed combinations (see scale a4", slide B43) and the feed (See adjoining side of duplex scale a47a47x.) yOn the upper or 4 power section A42 of this slide-rule is embodied the law of relations between the depth of the cut, (fixed scale a48,) the number of the tools taking the cut, (scale of broad arrows exponented l to 4 on slide B42) power of the material being cut, (scale a4, also on slide B42) the pressure available on the toolas the result of belt-pulling on a cone-step of lathe No. 43 of a certain diameter, (scale a0 of slide B42) through a certain back-gear 4 4 speed combination, (the, from the nature of the lathe No. 43, non-continuous scale am, eX- ponented A and B on slide B44,) and on toacertain diameter of work, the feed, (mechanically embodied in the adjoiningV or A41 side of the duplex scale w47.) This, the second equation of the problem for which the rule shown in Figs. 4, 5 is equipped, we, as indicated by the bracketed references, mechanically embody thereon. This slide-rule also comprises sections A41 A44, preferably united intoca single instrument in which the mechanical embodiments of its problems unknown variables when the rule is set merge into a common field of yvision whereby the rule becomes in any particular case an instrument apt to reveal its answer. The ultimate stage of the 'solution effected by this instrument, Figs. 4 and 5, may seem to be obtained not so directly as is the clear-cut solution of the problem illustrated as an example in connection with the slide-rule for helical springs, F-igs. l and 2, for it frequently happens that more than one speed and power combination will at the same time satisfy the requirement of utilizing, together with its coincident feed, both the full pulling power of the lathe and all there is of durability in the cutting tool or tools. However, the interdependence of feed and cutting speed under otherwise fixed conditions is such that of two or more exact solutions the one that implies the coarsestV feed will remove the most metal in a given time or, in other words, cause lathe No. 43 to remove a certain amount of metal from the selected work in a minimum of time. In addition the class number for IOO IIO

to this the lathe under consideration has in common with the rest of its class only a limited number both of feeds and of speed and power combinations, so that as a rule but e approximate solutiohs of the problem will ap- Such'a lathe as No. 43, being usual and IZO ISO

ditions concerning the cutting tool or tools of Vgo lathe No. 43, viz:

A. Upon section A, which is appropriate to the speed at which work is urged upon the tool, we select, say, first, tool to stand up to the Work for one hour,twenty minutes, (life of tool,) fixed scale az; second, a stream of water to be used on the tool, (with water,) arrow 1', slide B, but if dry set at arrow ro; third, a one-half-inch depth of cut to be taken, (set at one-half inch, scale a, slide B;) fourth, class No. 14 of material inwork, (set graduation 14, scale a, slide B2;) fifth, twentyinch diameter of work, (set twenty-inch, scale a on slide Bs.) Concerning the pulling power of the lathe, the rule upon section A is next set. B. Bycorrespondingmanipulations ofslides B and B to satisfy, of the above-given conditions, Nos. 3, 4, and 5, and besides by the appropriate setting of broad arrow 2, the condition that two cutting-tools are to be used at the same time. The instrument being thus set on both section A A, it first appears that there is no feed so fine that any of the speed and power combinations within the B gear reduction (see slide B, scale ame) would pull the two c uts, so we at once confine ourselves to the consideration of the A combinations. On doing this we readily discover that the combination 1 A F is the only one that appears nearly opposite to itself in both sections of the instrument; but unfortunately the t' feed is a triie too coarse both for the pulling power and the life of the tool, so that we are limited to the z. feed, which feed will neither utilize the full pulling power of the combination 1 A F nor the possibility of the cutting-tool in the vlife-limit put upon it. For this reason it at once becomes a question whether the combination 2 A F, which in the speed-section (A) of the instrument is seen to just coincide with the g feed, will not be more advantageous in spite of said g feed being finer than the L feed and the failure of this combination to,

vand l A F bear to each other. We thus have The ratio g being thus a triiie greater than 2 A F l A F, we realize that we gain more by using the slower combination 1 A F with the coarser feed L than by using the faster combination 2 A F with the finer feed g, while the gain in this instance is hardly a consideration. the same logarithmic scale are represented by equaldistances and equal ratios on two logarithmic scales of different dimensions are represented by proportional distances the arithmetical work done above may be performed by means of a pair of proportional dividers so set that its one end will give the same ratio on the scale of speed combinations a on the slide B3 that its other end will give on the scale of feeds. In fact, whenever the difference between the two ratios to be compared is a real item the comparison is readily made by the eye alone of the skilful manipulator, andwe are therefore justified in stating that this instrument, as well as the one shown in Figs. 1 and 2, reveals the answer.

The circular slide-rule, Figs. 12, 13, is to give answer to substantially the same general problem as the straight rule described in connection` with Figs.i 1, 2. The equations on which that instrument is based, therefore, are again mechanically embodied on this rule, the chief dierence being that conforming to the general plan of this modification the like scales @121 to a127 ment and only for the sake of simplicity less finely subdivided. This slide-rule is also, as

the ratio 4indicated by coincident heavy-faced lines '12 among its said scales, set to and gives on inspection of its self-indicatedfield of vision (see the concentrated similar groups of graduations upon the scales of the unknown variables thereofviz., of scales al and a1,) and, in the same manner, as first above explained in reference to the rule shown in Figs. 1, 2, reveals the answer. A

As to the mechanical detail of the rule, Figs. 12, 13, much will be familiar to those conversant with ordinary circular slide-rules, and therefore need not be described;4 but in addition it is proper here to point out that the amm) thereof are in circular arrange- However, as equal ratios on IOO IIO

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base or relatively fixed part of the instrument is the fiat disk A12", that in said bases center is transversely fixed as the center about which the relatively movable slides B121 B122 B123 swing a fiat-headed screw-bolt s. This bolt has engaged upon its screwed tip a combined sleeve-bolt and milled thumb-nut s. Of this nut the internal threads 8 engage and work along bolt s', thereby acting to bind or release, according to its turning, between said nut and the head of said bolt the base A20 and the lowest slide B123, a convenience to be availed of as soon as any two graduations, as i, on'

said parts are in any given case set, which being done the two upper slides-viz., sector B121 and disk B122-obviously may be swung to any designated position Without disturbance of the bound slide B123. When theywviz., B121 and B122-are so set in accordance with the particular problems terms, a thumb-nut s, engaged upon ascrew s, formed on the external body of nut '82, serves to bind the whole in place. Unscrewing of nuts s1 and s2 of course serves to release the parts on which they bear for resetting, &c., when such is desired. In order, compactly,to plot the (here) logarithmic scales expressive of the two equations incident to helical springs upon the rule shown in Figs. 12, 13, we found it desirable to depart somewhat froma strict sectional division of the instrument; but the several logarithmic scales going to make up such spring two distinct equations may, however, on inspection be clearly distinguished yfrom each other. ln order to see the scale e121 w12, if the disk B123, which overlies it, be of opaque material we form in disk B12s a sight-hole la.

' lf desirable, instead of having the two sections, as A A2, of any of the above-illustrated rules physically united in a single instrumentYA cooperative equation of the particularproblem whose answer is sought, then by setting the slides B112 B113 of that section to the vknown variables values to cause to appear thereon another set of relations between the unknown variables of the problem in hand, which being within otherllimits than those determined as to the same unknow variables by the A11 section of the rulesvn in Fig. 4) are useful to the computer for contrast with' the former,"and he, setting off the one against the other, soon acquires a facility in i discovering the answer common to both, which Inspection of the modication, Fig.`14, will show that it has upon an oblong base A110 twin centers of revolution, the pivots 811 811 v upon which are mounted in two sets for rotative sliding disk-like slides B112 B113 and B111 B115, upon one set of which and the convenient adjacent part A112 of base A111 aforesaid are embodied (the detail of which it would be tedious herein to relate) all but one of the two unknown variables of the two equations of lathe No. 43, and upon the other set of slides B111 B111 and their adjacent base part or section A112, the mechanical embodiment now too lfamiliar to repeat of the other equation. Between these two circular slides is embodied vupon base A110 the familiar split scale, (here lettered a1" a111x,) and dra-wn to it are directing-lines Z l', which lead the leye from those scales of the two circular slides which carry the other unknown variable common to the formulae or equation of lathe No. 43. This rule, Fig. 14, is also set to the identical problem of the rule described in connection with Figs. 4, 5, said setting being indicated by heavy-faced lines 110, coincident in adjoining scales, and the answer revealed similarly materials may be any customary or appropriate to slide rules of their several general shapes. f

The fact that the two equations thus embodied upon the above-described rules are not identical tends as we have contrived it in all the preferred self contained embodiments. (See not only Figs. 1 2, 4 5, but also Figs. 12, 13, and 14 of the illustrative examples of our invention for problems involving among their variables two unknown ones to a mutual delimitation)-that is to say, to a delimitation IOO IIO

by and between the ranges of result, which when any of the scales of said rules have been set to the requirements of a particular case are in their respective sectionsas A11 or A12, Fig. 4, carried to and made to appear in a common field ofvision, said common field of vision being that portion of the rule whereon after such setting similar' graduations of the relatively sliding' unknownvariable scales in Fig. 4, ar,11S w11, concentrate. Brought to suchr a conjunction, if the answer be not then reivealed by an exact coincidence among thegraduations of the adjoining unknown variable scales the selection of the answer from said field of vision still becomes so readily reducible as to be practically instantaneous to the skilled user. modification shown in Fig. 15, and thereby y extending our illustrative instruments to one dealing with more' than two unknown vari- Also before describing the i ables, it is as well here to recapitulate and consider the instruments now described as group. The underlying principle of this group of instruments is the development of a sliderule to solve that class of problems that involve two distinct or independent relations or equations between some or all of the variables of the problem. This end is in each said illustrative example accomplished by combining, as it were, two distinct slide-rules in one instrument, each rule (or as we have by reason of their merger into single instruments designated them veach rule-sections thereof) being the mechanical embodiment of one of these distinct sets of relations or equations existing between the variables. In general the middle relatively fixed portion of these instruments of dual components being the most convenient place carries the split or duplex scale, as c7 am, Fig. 1, representing one of the unknown variables, while on either flank thereof are the slides which carry to it, under the restrictive positions determined to them by the selected values of the known variable in any case, the other unknown variable, but there by reason ofY the difference in said equations in different and mutually delimiting scales. The two sets of distinct ratios ofthe two unknown variables of any case thus carried up and confronted in the rule are then so close together that the instrument in some instances (see Fig. 1) itself reveals at aglance in the coincidence of like graduations the answer; but if not by an absolute coincidence the observer in a similar occular way soon after and at a minimum of mechanical selection from the mutually delimiting and differently-dimensioned scales of the unknown variables discovers the answer.

In Fig. 15 is shown a slide-rule for the solution of a problem involving more than twoto wit, threeequations, and hence three unknown variables or quantities, the equations embodied on the instrument being (1.) z y I Z062 z I I) 3/ (3) wv I i which solved for c becomes (2.) e y m l z I c y.

The embodiment, Fig. 15, shows by coincident heavy lines 150, the solution of the equations for @29, 5:8, 0:2, 00:2 appearing opposite to the values of the variables a b c, and yz, opposite in the section, appears opposite to z: 12. The instrument consists in this case of three sections A1111 A, each being the embodiment of one of the aforesaid three equations, the sections A1 and A1" having the duplex z scale 0157 @157x in common, the third having the separate single .e scale L15/7 of otherwise precisely the same proportions. In this rule the three slides B11-1 B1G2 B153, being in shape and base engagement ordinary isolated T-section slides, need no descriptive text as to such detail. The adjusting of the slides B111 B152 B153 of the instrument to show the solution of the problem in the above-indicated manner is, however, a tentative process only, the method of procedure being iirst to try the value of unity (1) for by setting x21 opposite to the particular values of the case under consideration for the variables a, b, and c. By inspection we then readily see whether a: must be made greater or less than 1 in order to get coincident values of 1/ and e in line with each other in all three sections of the instrument, and by a few successive readjustments the solution is soon aiected. An instrument of this kind will be particularly useful in cases in which the unknown variables cannot have continuous values, but are confined to assume certain discontinuous values, for then a direct algebraic solution is impossible and a tentative algebraic solution very tedious.

In cases in which there are more than four variables in any of the three equations secondary slides carrying scales representing the additional variables can be introduced in an instrument of this kind in the same manner as in the instruments already described involving only two equations, (see Figs. 1, 2, 3, 4, and 5 to 11,) the variable in juxtaposition to the unknown variable a' being then the controlling one for this latter one.

Having now described our invention, what we claim as new, and desire to secure by Iletters Patent, is-

1. A slide-rule for the solution of a problem involving a number of unknown variables; said rule comprising a relatively iixed part divided into sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables en tering into the general problem of the rule, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown variables in their proper, mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfyl all the relations severally embodied in the dierent sections of the rule.

2. A slide-rule for the solution of a problem IOO IZO

involving a number of unknown variables; said rule comprising a relatively fixed part divided into sections, movable parts for these seetions, scales carried by these fixed and movable parts representing the variables entering` into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations, between the variables of the rule problem, or some of them, and coacting to locate the scales representing' two of the unknown variables in their proper, mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfy all the relations severally embodied in the different sections of the rule.

3. A slide-rule for the solution of a problem involving a number of unknown variables; said rule comprising a relatively xed part divided into sections, movable parts for these sec,- tions, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem and at least one of these represented by a scale in each section, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown variables in their proper, mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfy all the relations severally embodied in the different sections of the rule.

4L. A slide-rule for the solution of a problem involving a number of unknown variables; said rule comprising a relatively fixed part divided into sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem and one of these represented by a fixed scale in each section, and all the scales appertainingto each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown variables in their proper, mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfy all the relations severally embodied in the different sections of the rule.

5. A slide-rule for the solution of a problem involving a number of unknown variables; said rule comprising a relatively lixed part divided into sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem and one of these represented by similarly-located fixed scales of equal magnitudes, one in each section, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown variables in their proper, mutually-delin'iiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfy all the relations severally embodied in the different sections of the rule.

6. A slide-rule for the solution of a problem involving a number of unknown variables, said rule comprising a relatively fixed part divided into sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem and one of these represented by a fixed scale in each section and another by a movable scale in each section in juxtalocation to said fixed scale, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown 'variables in their proper, mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of the unknown variables which will satisfy all the relations severally embodied in the different sections of the rule.

7. A slide-rule for the solution of a problem involving a number of unknown variables; said rule comprising a relatively fixed part divided into sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, there being as many sections in the rule as there are unknown variables in the problem and one of these rep- IOO lIO

resented by similarly-located fixed scales of equal magnitudes, one in each section, and another by a movable scale in each section in juxtaloation l-to said fixed scale, and all the scales appertaining to each section constituting together an embodiment of one of the independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing two of the unknown variables in their proper mutually-delimiting relation to each other, the rule-sections being assembled in the rule as described, and so that the scales thus located in the several sections will again coact by inspection to determine the values of' the unknown variables which will satisfy all the relations severally embodied in the different Sections of the rule.

8. A slide-rule for the solution of a problem involving two unknown Variables; said rule comprising a relatively fixed part divided into 1 two sections, movable parts for these sections,

scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, and all the scales appertaining to each section constituting together an embodiment of one of the two independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing the two unknown variables in their proper, mutuallydelimiting relation to each other, the two rulesections being assembled in the rule as described, and so that the scales so located in the two sections will again coact by inspection to determine the values of the two .unknown variables which will satisfy both of the relations embodied in the two sections of the rule. 9. A slide-rule for the solution of a problem involving two unknown variables; said rule comprising a relatively fixed part divided into two sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, and one of the two unknown variables being represented by a fixed scale in each section, and all the scales appertaining to each section constituting together an embodiment of one of the two independent sets` of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing the two unknown variables in their proper mutually-delimiting relation to each. other, the two rule-sections being assembled. in the rule as described, and so that the scales so located in the two sections will again coact by inspection to determine the values of the two unknown variables which will satisfy both of the relations embodied in the two sections of the rule.

10. Aslide-rule for the solution ofaproblem involving two unknown variables; said rule comprising a relatively fixed part divided into two sections, movable parts for these sections,

scales cai ried by these fixed and movable parts representing the variables entering into the vgeneral problem of the rule, and one of the unknown variables being represented by similarly-located fixed scales of equal magnitudes, one in each section, and all the scales appertaining to each section constituting together an embodiment of one of the two independent sets of relations between the variables of the rule problem, Aor some of them,

and coacting to locate the scales representing` scales carried by these fixed and movable partsy representing the variables entering into the general problem of the rule, and one of the two unknown variables, beingi represented by a fixed scale in each section and the other by a movable scale in each section in juxtalocation to said dxed scale, and all the scales appertaining to each section constituting together an embodiment of' one of the two independent sets of relations between the variables of the rule problem, or some ofthem, and coacting to locate the scales representing the two unknown variables in their proper, mutually-delimiting relationto each other, the two rule-sections` being assembled in the rule as described, and so that the scales so located in the two sections will again coact by inspection to determine the values of the two unknown variables which will satisfy both of the relations embodied in the two sections of the rule.

12. A slide-rule for the solution of a problem involving two unknown variables, said rule comprising a relatively fixed part divided into IIO two sections, movable parts for these sections,

scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule, and one of the two unknown variables'being represented by similarly-located fixed scales of equal magnitudes, one in each section, and the other by a movable scale in each section in juxtalocation toV said fixed scale, and all the scales appertaining to each section constituting together an embodiment of one of the two independent sets of relations between the variables of the rule problem, or some of them, and coacting to locate the scales representing the two unknown variables in their proper, mutually-delimiting relation to each other, the two rule-sections being assembled in the rule as described, and so that the scales so located in the two sections will again coact by inspection to determine the values of the two unknown variables which will satisfy both of the relations embodied in the two sections of the rule.

13. A slide-rule for the solution of a problem involving two unknown variables; said rule comprising a relatively fixed part divided into two sections, movable parts for these sections, scales carried by these fixed and movable parts representing the variables entering into the general problem of the rule., and one of the two unknown variables being represented by one ixed scale common to the two sections, and the other by a movable scale in each section in juxtaiocation to said {ixed scale., and ali the scales appertaining to each section constituting together an embodiment of-ene of the two independent sets of relations between the variables of the rule problem, or some of them7 and coacting to locate the scales representing the two unknown variables in their proper, mu-

tually-delimiting relation to each other., the two rule-sections being assembled in the rule as described, and so that the scales so located in the two sections will again coact by inspection to determine the values of the two unknown variables which will satisfy both of the rellations embodied in the two sections of the ru e.

le. A slide-rule having relatively stationary and longitudinally-movable parts adapted to support and adjust the scales of the rule, and some of said parts recessed intermediate of their ends to receive detachable scale-strips, in combination with scale-strips adapted to t and interlock in said recessed portion.

CARL G. BARTH. HENRY L. GANTT. FRED. WV. TAYLUR. Witnesses:

CEAS. F. MYERS., D. STEWART. 

